Existence and uniqueness of reflecting diffusions in cusps

Costantini, Cristina; Kurtz, Thomas G.

doi:10.1214/18-ejp204

Cited by 10 publications

(21 citation statements)

References 20 publications

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“…Because of that our paper may be seen as an invitation for future analysis of the thin layer approximation in the context of stochastic processes. Perhaps stochastic analysis, and the results presented in [28,58] in particular, may lead to generalizations that are also meaningful in this field.…”

Section: Comparison With the Existing Literaturementioning

confidence: 98%

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski

2020

J. Evol. Equ.

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Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

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Section: Comparison With the Existing Literaturementioning

confidence: 98%

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski

2020

J. Evol. Equ.

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“…Clearly, every solution of the constrained martingale problem is also a solution of the submartingale problem. This approach, or the corresponding one for stochastic equations, has been used, for example, in [10,5,7]. Whether the submartingale problem approach or the constrained martingale problem approach is used, the critical issue is uniqueness of the solution, which is still an open question for many examples (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Markov selection for constrained martingale problems

Costantini¹,

Kurtz²

2019

Electron. J. Probab.

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Constrained Markov processes, such as reflecting diffusions, behave as an unconstrained process in the interior of a domain but upon reaching the boundary are controlled in some way so that they do not leave the closure of the domain. In this paper, the behavior in the interior is specified by a generator of a Markov process, and the constraints are specified by a controlled generator. Together, the generators define a constrained martingale problem. The desired constrained processes are constructed by first solving a simpler controlled martingale problem and then obtaining the desired process as a time-change of the controlled process.As for ordinary martingale problems, it is rarely obvious that the process constructed in this manner is unique. The primary goal of the paper is to show that from among the processes constructed in this way one can "select", in the sense of Krylov, a strong Markov process. Corollaries to these constructions include the observation that uniqueness among strong Markov solutions implies uniqueness among all solutions.These results provide useful tools for proving uniqueness for constrained processes including reflecting diffusions.The constructions also yield viscosity semisolutions of the resolvent equation and, if uniqueness holds, a viscosity solution, without proving a comparison principle.We illustrate our results by applying them to reflecting diffusions in piecewise smooth domains. We prove existence of a strong Markov solution to the SDE with reflection, under conditions more general than in [13]: In fact our conditions are known to be optimal in the case of simple, convex polyhedrons with constant direction of reflection on each face ([10]). We also indicate how the results can be applied to processes with Wentzell boundary conditions and nonlocal boundary conditions.

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“…The assertion follows essentially from Condition 3.3(iv). The proof is analogous to that of Lemma 4.2 of Costantini and Kurtz (2018) and Lemma 6.4 of Taylor and Williams (1993).…”

Section: Formulation Of the Problem And Preliminariesmentioning

confidence: 88%

“…In particular, we replace the application of the Krein-Rutman theorem in Kwon and Williams (1991) by the inhomogeneous ergodic theorem of Section 2. In turn, some of the estimates we need to apply our probabilistic results exploit some analytical results of Kwon and Williams (1991) and a result proved in Appendix A1, together with the coupling result of Lemma 5.3 of Costantini and Kurtz (2018). Another essential ingredient of our arguments is that, in order to prove uniqueness of the solution of the constrained martingale problem for (A, D, B, Ξ), it is enough to prove uniqueness among strong Markov, natural solutions (Costantini and Kurtz (2019)).…”

Section: Outline Of the Proof Of Uniquenessmentioning

confidence: 99%

“…In order to verify the bound (ii), we use a coupling argument based on Lemma 5.3 of Costantini and Kurtz (2018) (Lemma 3.33), the fact that, for any reflecting diffusion, X, the rescaled process 2 2n X(2 −4n •) converges to the reflecting Brownian motion in the cone (the above mentioned Lemma 3.32), and the support theorem of Kwon and Williams (1991).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions

Costantini¹,

Kurtz²

2021

Preprint

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Bass and Pardoux (1987) deduce from the Krein-Rutman theorem a reverse ergodic theorem for a sub-probability transition function, which turns out to be a key tool in proving uniqueness of Reflecting Brownian Motion in cones in Kwon and Williams (1991) and in Taylor and Williams (1993). By a different approach, we are able to prove an analogous reverse ergodic theorem for a family of inhomogeneous sub-probability transition functions. This allows us to prove existence and uniqueness for a diffusion process with varying, oblique reflection, in a domain with one singular point that can be approximated, near the singular point, by a smooth cone, under natural, easily verifiable geometric conditions.

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Existence and uniqueness of reflecting diffusions in cusps

Cited by 10 publications

References 20 publications

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Markov selection for constrained martingale problems

A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions

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